Mixed-integer programming approaches for some non-convex and combinatorial optimization problems.

摘要

In this dissertation we study several nonconvex and combinatorial optimization problems with applications in production planning, machine learning, advertising, statistics, and computer vision. The common theme is the use of algorithmic and modelling techniques from mixed-integer programming (MIP) which include formulation strengthening, decomposition, and linear programming (LP) rounding.;We first consider MIP formulations for piecewise linear functions (PLFs) that are evaluated when an indicator variable is turned on. We describe modifications to standard MIP formulations for PLFs with desirable theoretical properties and superior computational performance in this context.;Next, we consider a production planning problem where the production process creates a mixture of desirable products and undesirable byproducts. In this production process, at any point in time the fraction of the mixture that is an undesirable byproduct increases monotonically as a function of the cumulative mixture production up to that time. The mathematical formulation of this continuous-time problem is nonconvex. We present a discrete time mixed-integer nonlinear programming (MINLP) formulation that exploits the increasing nature of the byproduct ratio function. We demonstrate that this new formulation is more accurate than a previously proposed MINLP formulation. We describe three different mixedinteger linear programming (MIP) approximation and relaxation models of this nonconvex MINLP, and derive modifications that strengthen the LP-relaxations of these models. We provide computational experiments that demonstrate that the proposed formulation is more accurate than the previous formulation, and that the strengthened MIP approximation and relaxation models can be used to obtain near-optimal solutions for large instances of this nonconvex MINLP.;We then study production planning problems in the presence of realistic business rules like taxes, tariffs, and royalties. We propose two different solution techniques. The first is a MIP formulation while the second is a search algorithm based on a novel continuous domain formulation. We then discuss decomposition methods to compute bounds on the optimal solution. Our computational experiments demonstrate the impact of our formulations, solution techniques, and algorithms on a sample application problem.;Finally, we study three classes of combinatorial optimization problems: set packing, set covering, and multiway-cut. Near-optimal solutions of these combinatorial problems can be computed by rounding the solution of an LP. We show that one can recover solutions of comparable quality by rounding an approximate LP solution instead of an exact one. These approximate LP solutions can be computed efficiently by solving a quadratic-penalty formulation of the LP using a parallel stochastic coordinate descent method. We derive worst-case runtime and solution quality guarantees of this scheme using novel perturbation and convergence analyses. Our experiments demonstrate that on these combinatorial problems our rounding scheme is up to an order of magnitude faster than Cplex (a commercial LP solver) while producing solutions of similar quality.